"Frontmatter". In: Analysis of Financial Time Series

132 NONLINEAR TIME SERIESr(t)-20 -10 0 100 2000 4000 6000 8000Figure 4.2. Time plot of the daily log returns for IBM stock from July 3, 1962 to December31, 1999.daysmodels of Chapter 3 are entertained, we obtain the following AR(2)-GARCH(1, 1)model for the seriesr t = 0.067 − 0.023r t−2 + a t ,a t = σ t ɛ tσ 2t = 0.031 + 0.076a 2 t−1 + 0.915σ 2t−1 , (4.10)where r t is the log return, {ɛ t } is a Gaussian white noise sequence with mean zero andvariance 1.0, the standard errors of the parameters in the mean equation are 0.013and 0.011, respectively, and those of the volatility equation are 0.003, 0.002, and0.003, respectively. All estimates but the coefficient of r t−2 are highly significant.The Ljung–Box statistics of the standardized residuals give Q(10) = 11.31(0.33)and Q(20) = 27.00(0.14), where the number in parentheses denotes p value.For the squared standardized residuals, we obtain Q(10) = 11.86(0.29) andQ(20) = 19.19(0.51). The model is adequate in modeling the serial dependenceand conditional heteroscedasticity of the data. But the unconditional mean for r t ofmodel (4.10) is 0.065, which is substantially larger than the sample mean 0.045,indicating that the model might be misspecified. The TAR model can be used torefine the model by allowing for asymmetric response in volatility to the sign ofshock a t−1 . More specifically, we consider an AR(2)-TAR-GARCH(1, 1) model for